Math Problem Statement

The line joining the points with coordinates (q, 2 - q) and (2q + 2, q - 4) is perpendicular to the line with equation 3x - 4y + 5 = 0. What is the value of q?

Solution

We are given two points with coordinates (q,2q)(q, 2 - q) and (2q+2,q4)(2q + 2, q - 4), and the line joining these points is perpendicular to the line 3x4y+5=03x - 4y + 5 = 0.

To find the value of qq, let's go through the following steps:

Step 1: Find the slope of the given line 3x4y+5=03x - 4y + 5 = 0.

We need to rewrite this equation in slope-intercept form y=mx+by = mx + b, where mm is the slope:

3x4y+5=04y=3x5y=34x+543x - 4y + 5 = 0 \quad \Rightarrow \quad -4y = -3x - 5 \quad \Rightarrow \quad y = \frac{3}{4}x + \frac{5}{4}

So, the slope of the given line is 34\frac{3}{4}.

Step 2: Find the slope of the line joining the points (q,2q)(q, 2 - q) and (2q+2,q4)(2q + 2, q - 4).

The formula for the slope of a line between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is:

slope=y2y1x2x1\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}

Substitute the coordinates of the points:

slope=(q4)(2q)(2q+2)q=q42+q2q+2q\text{slope} = \frac{(q - 4) - (2 - q)}{(2q + 2) - q} = \frac{q - 4 - 2 + q}{2q + 2 - q} slope=2q6q+2\text{slope} = \frac{2q - 6}{q + 2}

Step 3: Use the perpendicularity condition.

If two lines are perpendicular, the product of their slopes is 1-1. Let the slope of the given line be 34\frac{3}{4} and the slope of the line joining the points be 2q6q+2\frac{2q - 6}{q + 2}. Set up the equation:

34×2q6q+2=1\frac{3}{4} \times \frac{2q - 6}{q + 2} = -1

Step 4: Solve for qq.

Now, solve the equation:

3(2q6)4(q+2)=1\frac{3(2q - 6)}{4(q + 2)} = -1

Multiply both sides by 4(q+2)4(q + 2) to eliminate the denominator:

3(2q6)=4(q+2)3(2q - 6) = -4(q + 2)

Expand both sides:

6q18=4q86q - 18 = -4q - 8

Move all terms involving qq to one side and constants to the other side:

6q+4q=1886q + 4q = 18 - 8 10q=1010q = 10

Solve for qq:

q=1q = 1

Final Answer:

The value of qq is 11, which corresponds to option C.

Would you like further details on this process or additional explanations?

Here are five related questions you might want to explore:

  1. How do you find the equation of a line perpendicular to a given one?
  2. What is the general condition for two lines to be perpendicular?
  3. How do you calculate the midpoint between two points in coordinate geometry?
  4. Can you find the equation of a line passing through two points?
  5. What happens when two lines are parallel?

Tip: When working with slopes, always remember that perpendicular slopes are negative reciprocals of each other!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Coordinate Geometry
Slopes of Perpendicular Lines
Linear Equations

Formulas

Slope formula: (y2 - y1) / (x2 - x1)
Perpendicular slope condition: m1 * m2 = -1

Theorems

Perpendicular Slopes Theorem

Suitable Grade Level

Grades 10-12